Circular cut codes in genetic information

In this work we present an analysis of the dinucleotide occurrences in the three codon sites 1–2, 2–3 and 1–3, based on a computation of the codon usage of three large sets of bacterial, archaeal and eukaryotic genes using the same method that identified a maximal $C^3$ self-complementary trinucleotide circular code $X$ in genes of bacteria and eukaryotes in 1996 (Arquès and Michel, 1996). Surprisingly, two dinucleotide circular codes are identified in the codon sites 1–2 and 2–3. Furthermore, these two codes are shifted versions of each other. Moreover, the dinucleotide code in the codon site 1–3 is circular, self-complementary and contained in the projection of $X$ onto the 1st and 3rd bases, i.e. by cutting the middle base in each codon of $X$. We prove several results showing that the circularity and the self-complementarity of trinucleotide codes is induced by the circularity and the self-complementarity of its dinucleotide cut codes. Finally, we present several evolutionary approaches for an emergence of trinucleotide codes from dinucleotide codes.